# An Introduction to Elliptic Curves

Elliptic Curves and the Abelian Grou

Elliptic Curves are curves of the form

$y^2=x^3+ax+b$

This is called a

**Weierstrass equation**. The rational points on this elliptic curve miraculously create an**abelian group**under an operation called point addition.Given two points

$P$

and $Q$

, we can calculate a third point $P \oplus Q$

on the elliptic curve itself using a simple algorithm:The images here are from

**Joseph Silverman**'s presentation slides from 2006, which are an excellent reference and can be found here.

First, we draw the line

$L$

through the points $P$

and $Q$

, which intersects the curve at a third point $R$

. We then draw a vertical line passing through $R$

which intersects the curve at a second point we call $R^\prime$

. $R^\prime$

is the result of $P \oplus Q$

, the point addition of $P$

and $Q$

.We can also add a point

$P$

to itself, but how can we do that with infinitely many lines passing through? We simply take the tangent to the curve and find the point $R$

it intersects at before mirroring the point to $R^\prime = 2P$

.Rarely, you may find yourself adding

$P$

to $-P$

, the point directly below $P$

. In this case there is no third point of intersection and we say the result of point addition is $\mathcal{O}$

, the point "at infinity".Last modified 1yr ago